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Midwestern State University Minimum Spanning Trees Definition of MST Generic MST algorithm Kruskal's algorithm Prim's algorithm 1.

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Presentation on theme: "Midwestern State University Minimum Spanning Trees Definition of MST Generic MST algorithm Kruskal's algorithm Prim's algorithm 1."— Presentation transcript:

1 Midwestern State University Minimum Spanning Trees Definition of MST Generic MST algorithm Kruskal's algorithm Prim's algorithm 1

2 2 Definition of MST Let G=(V,E) be a connected, undirected graph. For each edge (u,v) in E, we have a weight w(u,v) specifying the cost (length of edge) to connect u and v. We wish to find a (acyclic) subset T of E that connects all of the vertices in V and whose total weight is minimized. Since the total weight is minimized, the subset T must be acyclic (no circuit). Thus, T is a tree. We call it a spanning tree. The problem of determining the tree T is called the minimum-spanning-tree problem.

3 3 Application of MST: an example In the design of electronic circuitry, it is often necessary to make a set of pins electrically equivalent by wiring them together. Running cable TV to a set of houses. What’s the least amount of cable needed to still connect all the houses?

4 4 Here is an example of a connected graph and its minimum spanning tree: a b h cd e fg i 4 87 9 10 14 4 2 2 6 1 7 11 8 Notice that the tree is not unique: replacing (b,c) with (a,h) yields another spanning tree with the same minimum weight.

5 5 Growing a MST Set A is always a subset of some minimum spanning tree.. An edge (u,v) is a safe edge for A if by adding (u,v) to the subset A, we still have a minimum spanning tree. GENERIC_MST(G,w) 1A:={} 2while A does not form a spanning tree do 3find an edge (u,v) that is safe for A 4A:=A ∪ {(u,v)} 5return A

6 6 How to find a safe edge We need some definitions and a theorem. A cut (S,V-S) of an undirected graph G=(V,E) is a partition of V. An edge crosses the cut (S,V-S) if one of its endpoints is in S and the other is in V-S. An edge is a light edge crossing a cut if its weight is the minimum of any edge crossing the cut.

7 7 V-S↓ a b h cd e fg i 4 87 9 10 14 4 2 2 6 1 7 11 8 S↑S↑ ↑ S ↓ V-S This figure shows a cut (S,V-S) of the graph. The edge (d,c) is the unique light edge crossing the cut.

8 8 The algorithms of Kruskal and Prim The two algorithms are elaborations of the generic algorithm. They each use a specific rule to determine a safe edge in the GENERIC_MST. In Kruskal's algorithm, –The set A is a forest. –The safe edge added to A is always a least-weight edge in the graph that connects two distinct components. In Prim's algorithm, –The set A forms a single tree. –The safe edge added to A is always a least-weight edge connecting the tree to a vertex not in the tree.

9 9 Kruskal's algorithm (simple) (Sort the edges in an increasing order) A:={} while (E is not empty do) { take an edge (u, v) that is shortest in E and delete it from E If (u and v are in different components)then add (u, v) to A } Note: each time a shortest edge in E is considered.

10 10 Kruskal's algorithm 1 function Kruskal(G = : graph; length: A → R+): set of edges 2 Define an elementary cluster C(v) ← {v}. 3 Initialize a priority queue Q to contain all edges in G, using the weights as keys. 4 Define a forest T ← Ø //T will ultimately contain the edges of the MST 5 // n is total number of vertices 6 while T has fewer than n-1 edges do 7 // edge u,v is the minimum weighted route from u to v 8 (u,v) ← Q.removeMin() 9 // prevent cycles in T. add u,v only if T does not already contain a path // between u and v. 10 // the vertices has been added to the tree. 11 Let C(v) be the cluster containing v, and let C(u) be the cluster containing u. 13 if C(v) ≠ C(u) then 14 Add edge (v,u) to T. 15 Merge C(v) and C(u) into one cluster, that is, union C(v) and C(u). 16 return tree T

11 11 http://Wikipedia/kruskals This is our original graph. The numbers near the arcs indicate their weight. None of the arcs are highlighted. Kruskal's algorithm

12 12 http://Wikipedia/kruskals AD and CE are the shortest arcs, with length 5, and AD has been arbitrarily chosen, so it is highlighted. Kruskal's algorithm

13 13 http://Wikipedia/kruskals CE is now the shortest arc that does not form a cycle, with length 5, so it is highlighted as the second arc. Kruskal's algorithm

14 14 http://Wikipedia/kruskals The next arc, DF with length 6, is highlighted using much the same method. Kruskal's algorithm

15 15 http://Wikipedia/kruskals The next-shortest arcs are AB and BE, both with length 7. AB is chosen arbitrarily, and is highlighted. The arc BD has been highlighted in red, because there already exists a path (in green) between B and D, so it would form a cycle (ABD) if it were chosen. Kruskal's algorithm

16 16 http://Wikipedia/kruskals The process continues to highlight the next-smallest arc, BE with length 7. Many more arcs are highlighted in red at this stage: BC because it would form the loop BCE, DE because it would form the loop DEBA, and FE because it would form FEBAD. Kruskal's algorithm

17 17 http://Wikipedia/kruskals Finally, the process finishes with the arc EG of length 9, and the minimum spanning tree is found. Kruskal's algorithm

18 18 Prim's algorithm (simple) MST_PRIM(G,w,r){ A={} S:={r} (r is an arbitrary node in V) Q=V-{r}; while Q is not empty do { take an edge (u, v) such that u  S and v  Q (v  S ) and (u,v) is the shortest edge add (u, v) to A, add v to S and delete v from Q }

19 19 Prim's algorithm for each vertex in graph set min_distance of vertex to ∞ set parent of vertex to null set minimum_adjacency_list of vertex to empty list set is_in_Q of vertex to true set min_distance of initial vertex to zero add to minimum-heap Q all vertices in graph, keyed by min_distance Initialization inputs: A graph, a function returning edge weights weight-function, and an initial vertex Initial placement of all vertices in the 'not yet seen' set, set initial vertex to be added to the tree, and place all vertices in a min-heap to allow for removal of the min distance from the minimum graph. Wikipedia

20 20 Prim's algorithm while latest_addition = remove minimum in Q set is_in_Q of latest_addition to false add latest_addition to (minimum_adjacency_list of (parent of latest_addition)) add (parent of latest_addition) to (minimum_adjacency_list of latest_addition) for each adjacent of latest_addition if ((is_in_Q of adjacent){ and (weight-function(latest_addition, adjacent) < min_distance of adjacent)) set parent of adjacent to latest_addition set min_distance of adjacent to weight-function(latest_addition, adjacent) } update adjacent in Q, order by min_distance Wikipedia The while loop will fail when remove minimum returns null. The adjacency list is set to allow a directional graph to be returned. time complexity: V for loop, log(V) for the remove function time complexity: E/V, the average number of vertices time complexity: log(V), the height of the heap

21 21 Grow the minimum spanning tree from the root vertex r. Q is a priority queue, holding all vertices that are not in the tree now. key[v] is the minimum weight of any edge connecting v to a vertex in the tree. parent[v] names the parent of v in the tree. When the algorithm terminates, Q is empty; the minimum spanning tree A for G is thus A={(v,parent[v]):v ∈ V-{r}}. Running time: O(||E||lg |V|). Prim's algorithm

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